11.7 Day 1 Cylindrical Coordinates

Comparing Cartesian and cylindrical coordinates

Note: these are just polar coordinates with a z coordinate (z is a vertical component)

Conversion formulas from cylindrical to rectangular

coordinates

Converting between cylindrical coordinates and

rectangular (Cartesian)Note: these formulas must be memorized

Example 1

Convert the point (r, ө, z) = (4, 5π/6, 3)

to rectangular coordinates.

Solution to example 1

Example 2

_

Convert the point (x, y, z) = (1, √3 , 2)

to cylindrical coordinates.

Cylindrical coordinates are usually more convenient for representing cylindrical surfaces as they often result in simpler

equations.

Vertical planes containing the z-axis and horizontal planes also have simple

cylindrical coordinate equations

Example 3 a

Find an equation in cylindrical coordinates for the surfaces represented by the rectangular equation:

Solution to 3aFrom the preceding section, you know that x2 + y2 = 4z2

is a “double napped” cone with its axis along the z-axis as shown.

If you replace x2 + y2 with r2, the equation in cylindricalCoordinates is r2 = 4z2

x2 + y2 = 4z2 Rectangular equation

r2 = 4z2 Cylindrical equation

Example 3b

Find an equation in cylindrical coordinates for the surfaces represented by the rectangular equation:

Solution to 3b

Example 4

Find an equation in rectangular coordinates for the surface represented by the cylindrical equation:

Identify the surface

r2cos2θ +z2 +1 = 0

y

z

x

Changing between coordinates on the TI 89

Press 2nd 5 (math) – 4 matrices – L Vecor ops

To polar, to cynd To convert rectangular to Polar (2 D) or cylindrical (3D)

[1,2] to Polar (to expressed with a triangle)

[1,2,3] to Cylind

Bonus material on the next slides

This is a polar bear in rectangular form

Note: Homework do the assignment sheet plus the activity on the class website.

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